Impossibility, Subjective Probability, and Punishment for
Attempts

Stephen Shavell, in his recent article "Deterrence and the
Punishment of Attempts," raises the question of whether people should
be punished for attempted crimes that are impossible--have zero
probability of succeeding. Two examples are the person who attempts
to commit murder by voodoo and the pickpocket who attempts to pick an
empty pocket. His conclusion is that "we can reformulate the idea of
"impossible" attempts so that it corresponds to those acts that carry
with them a negligible probability of harm. Such acts should not be
punished because there is no need to deter them."

In his analysis, Shavell considers the objective probability that
an act will succeed and the (subjective) probabilities held by the
court. He does not consider probabilities from the standpoint of the
offender. The purpose of this note is to argue that it is crucial, in
analyzing the question of punishment for impossible attempts, to
consider the subjective probabilities of the offender--his beliefs as
to what methods of committing the crime work.

Obviously, if the offender knows that the method he is using will
not work, he will not use it--or, if he does, he is not in fact
attempting to commit the crime. The important case is the one where
the court knows that the method is impossible but the offender does
not. If the offender is aware of his own ignorance and rationally
allows for it in his decision, then, as we will see, punishing
impossible attempts does in fact deter offenses. If, on the other
hand, the offender believes that he has perfect knowledge about what
methods work, but is wrong, then punishing impossible attempts serves
no function. We will start with the former case.

Rational Voodoo

Consider the following simple example. I am considering committing
a murder in one of two ways--poison or voodoo. The poison I am
considering is invariably lethal, whereas sticking pins in a voodoo
doll will have no effect at all on the prospective victim's life
expectancy.

If I were aware of these facts, I would either choose poison or
not attempt the murder at all. The problem arises because I am not
aware of them. I know that the methods I am contemplating may not
work, but I do not know which is more likely to work. Assume, for
simplicity, that I know one of the methods works with ps and the
other has no effect, but I have no idea which is which.

Shavell's analysis is based on the deterrent effect of punishment,
although he also discusses other functions, such as putting a
criminal in jail in order to make future crimes impossible. My
analysis will also be based on deterrence. To entirely eliminate the
issue of incapacitation, I assume that the criminal will make at most
one attempt; having spent his savings on either poison or a voodoo
doll and failed, he will give up.

Like Shavell, I assume that, whichever method is used, there is
some probability that the offender will be detected before the crime
is complete. The question is whether, if he is detected, he should be
punished only if he is using a method that might work. Shavell's
answer is yes, mine is no.

The crucial point is that, from the criminal's standpoint, the
legal rule is: Attempts by impossible means are (or are not)
punishable. Since the criminal does not know which method is
impossible (if he did, he would not bother using it), this does not
translate, for him, into attempts by voodoo are (or are not)
punishable. Since he knows that the method he is planning to use
might turn out to be impossible, a policy of punishing attempts that
(turn out to be) impossible is to raise the ex-ante cost to him of
trying to commit murder.

The argument can be made more formal as follows:

Let pa be the probability that an attempt (of either
sort) is detected.

Let pc be the probability that an offense (in this case
murder) is detected.

Let Fi be the punishment for an impossible attempt.

Let Fp be the punishment for a possible attempt.

Let Fc be the punishment for committing an offense.

Let ps be the probability that a possible attempt will
succeed.

In the example given, the expected punishment (as perceived by the
offender) for attempting to commit murder is:

pa(Fi+Fp)/2 +
(1/2)pspcFc

Here the first term is the probability that the attempt will be
detected times the resulting expected punishment--a fifty percent
chance that the method turns out to be impossible (punishment
Fi) plus a fifty percent chance that it turns out to be
possible (Fp). The second term is a fifty percent chance
that the method chosen is correct, multiplied by the probability that
the correct method will work, multiplied by the expected punishment
for an offense. As you can see from equation 1, a punishment for an
impossible offense is, in this case at least, just as effective a
deterrent as a punishment for a possible offense. If, as Shavell
assumes, punishment is costless but has an upper bound, we are
indifferent among any combinations of punishments for possible and
impossible attempts having a given sum, and will punish both kinds of
offenses if the desired total is too high to be achieved by punishing
only one. If, more realistically, marginal punishment cost increases
with the size of
punishment,[1] we
produce a given level of deterrence at the lowest cost by imposing
equal punishments for both the possible and the impossible attempt.

So far, I have assumed that the offender knows nothing at all
about which method is impossible. Consider, at the opposite extreme,
an offender who knows perfectly well which method works; if he
chooses to use the method that does not it is in order to achieve
some objective other than committing the crime (perhaps he wants to
impress a friend who believes in voodoo). In this case, punishment
for the impossible attempt has no tendency to prevent the offense,
but does tend to prevent the (harmless) attempt, hence Shavell's
conclusion is correct.

Finally, consider an offender who is certain that the impossible
method is the one that works. The knowledge that impossible attempts,
if detected, will be punished has no deterrent effect on him, since
he is certain the method he is using is not the impossible one.

The argument can be expanded to cover the more general case where
the offender has some, but imperfect, knowledge about the
effectiveness of the alternative methods. This is done in the
appendix, where we analyze the case of a potential offender whose
subjective probability that the method that is actually impossible
(voodoo) is the possible one is p. The conclusion is that if p<.5
(the offender considers the method that in fact is impossible less
likely to work than the method that in fact is possible), then
punishing impossible attempts deters offenses, although a given level
of punishment of impossible attempts provides less deterrence than an
equal punishment of possible attempts. If the offender's beliefs are
perverse--he thinks voodoo more likely to work than poison--he will
either not attempt to commit the offense or use the impossible
method.

Irrational Voodoo and the Reasonable Man

Suppose we believe that there are two kinds of people. Reasonable
people know which methods of committing crimes are impossible and do
not employ them (p=0). Unreasonable people sometimes employ
impossible methods. Unreasonable people do not realize that they are
unreasonable; they believe that they know which methods are
impossible (p=1). The knowledge that impossible attempts will be
punished does not increase their (subjective) expected punishment,
hence does not deter them. So there is no reason to punish impossible
attempts. We are back with Shavell's result.

This argument may justify defining "impossibility" by a reasonable
man standard: An attempt is impossible if a reasonable man would know
that it would not work. An attempt that is in fact impossible, but
that a reasonable man might believe is possible (trying to pick a
pocket that in fact is empty) should still be punished, as per our
earlier analysis.

To put the argument differently, a method that a reasonable man
would regard as impossible is one for which p has a sharply bipolar
distribution. Almost everyone either is reasonable and is sure voodoo
does not work (p=0) or is unreasonable and thinks it probably does
work (p>.5). Neither group will be deterred from murder by the
knowledge that impossible attempts are punished, so there is no
reason to punish them. A method that a reasonable man would regard as
possible, on the other hand, is one for which many potential
offenders will have 0<p<.5, hence punishing attempts of that
sort that turn out to be impossible provides useful deterrence.

The Many Person Case

Our argument so far involves deterring a single offense by a
single offender. The next stage is to consider a population of
potential offenders, with varying values of p. Those for whom p>.5
will either not attempt the offense or pick the impossible method;
they will produce no offenses but may produce some convictions for
(impossible) attempts. Those for whom p<.5 will either not attempt
the offense or choose the correct (possible) method. As can be seen
from Equation 4 of the appendix, the knowledge that impossible
attempts, if detected, will be punished increases the expected cost
of the offense. The size of the effect is proportional to p/(1-p),
hence increases with increasing p. So a policy of punishing
impossible attempts selectively deters those for whom p is relatively
large--potential offenders who are uncertain as to which method
works.

The argument may be made verbally as well as mathematically. If
you are unsure which method works you must allow in your calculations
for the chance you will choose the wrong one. If impossible attempts
are not punished, then the wrong choice means that you will not
succeed in your crime but will also not be punished. If impossible
attempts are punished, you risk using an impossible method and being
punished for your attempt. That possibility is one of the costs you
must take into account in deciding whether or not to attempt the
offense. The potential offender who is sure he knows which method
works does not have to worry about that problem.

So a policy of punishing impossible attempts tends to deter
offenses, especially by those offenders uncertain as to which method
works. The cost of that deterrence is that some people caught making
impossible attempts must be punished. These will, by the analysis of
the appendix, be people for whom p>.5--potential offenders who
believe that voodoo is more effective than poison. The offenses that
will be deterred will be by potential offenders who think poison is
more effective than voodoo, but are not sure. Whether punishing
impossible attempts is a relatively efficient or inefficient way of
deterring crime will depend on the relative sizes of the two
groups.[2] Even if
punishing impossible attempts has a relatively small deterrent effect
(because most potential offenders have p<<.5), it may be an
efficient form of deterrence if there are very few impossible
attempts that need be punished (because very few potential offenders
have p>.5).[3]

Are All Failed Attempts Irrational?

One objection to both my analysis and Shavell's might be that the
distinction we make between "possible" and "impossible" attemts is
meaningless. All unsuccessful attempts are impossible ex post;
they differ only in the degree to which the perpetrator knew that
they were impossible. The only possible attempts are those that
succeed.

It is useful, and reasonably straightforward, to reformulate my
analysis in these terms. In the simplest case, we have two kinds of
methods--those that work with certainty and those that fail. The
question of whether to punish impossible attempts becomes the
question of whether to punish unsuccessful attempts. The answer, as
shown above, depends on the distribution of subjective probabilities
in the population of potential offenders. The analysis is the same as
before, with ps=1 and Fp=0.

More generally, we have a large number of methods, some of which
work with certainty and some of which fail. Among the failed methods,
some are ones for which the perpetrator had a high subjective
probability of success (putting cyanide in a glass of water and
handing it to the victim--who drops the glass) and some ones for
which the perpetrator had a low subjective probability of success.
Applying our analysis to this more general case, with a distribution
of subjective probabilities over both methods and potential
perpetrators, would tell us which impossible (i.e. failed) attempts
should be punished and which labelled "impossible" and excused.

Irrationality

So far I have assumed that potential offenders are rational and
fully informed about everything affecting their crime except what
method works--that even people who believe in voodoo know the legal
doctrine with regard to impossible attempts and correctly allow for
its effect on their expected punishment. If this is not true, the
results of punishing impossible attempts are much less clear; they
depend on precisely who is how irrational or poorly informed, and in
what way.

Suppose, for instance, that anyone who thinks voodoo might work is
so badly informed, or so irrational, that actual legal doctrines have
no effect on him at all. Such people may make impossible attempts and
be punished for them, but they will not be deterred by that
possibility since they will not be aware of it. In that case,
punishment of impossible attempts will have no deterrent effect.

How plausible we think this argument is depends partly on how
obvious it is which methods are impossible; someone may be both
rational and well informed about the law, yet still try to pick a
pocket that happens to be empty. It also depends on how we believe
that poorly informed and/or irrational people make decisions. If, for
instance, irrationality takes the form of overestimating the
probability of unlikely events, a claim sometimes used to explain the
purchase of lottery tickets, then it may increase deterrence--if, as
seems likely, the apprehension of someone using voodoo to try to
commit murder is an unlikely event.

Incapacitation

Suppose we expand our analysis to allow for incapacitation as a
purpose of the penal system. We might then argue that people who
attempt impossible means are harmless; their knowledge of real-world
causality is sufficiently bad that anything they attempt, including
murder, is likely to fail. Hence there is no advantage to imprisoning
them.

This argument can, however, be made the other way. One of the
things we learn from a correct knowledge of causality (at least under
an efficient legal system) is that killing people may well cause us
to be punished. People who do not understand causality may fail to
perceive that relation. If so, they will attempt more murders than
would more rational people--and eventually they may give up on voodoo
and use poison instead.

Conclusion

In the course of this note I have tried to establish two
conclusions, one substantive and one methodological. The former is
that punishing impossible attempts provides some deterrence against
actual crimes, and may even be an efficient way of deterring them.
The latter is that, in analysing issues of deterrence, we must take
into account the beliefs of the people we are trying to deter, as
embodied in their subjective probabilities, since it is on the basis
of those beliefs that they will act.

Looking at Equation 3, we observe that the right hand side does
not depend on Fp and Fi separately, but only on
their sum. If our objective is to make method 2 more attractive than
method 1, we can do so by making the punishments sufficiently large
(if p<.5) or small (if p>.5), but we need not distinguish
between possible and impossible attempts. Note that p<.5
corresponds to the (more plausible) assumption: the offender does not
know which method works, but he considers the possible method more
likely to work than the impossible method.

Under what circumstances will the offender choose method 2 over no
attempt at all? From Eqn. 2 we have:

Comparing this with Equation 3, we observe that, if p<.5 and
the rhs of Eqn. 3 is negative, then the rhs of Equation 4 is negative
a fortiori, so the equation does not hold. Hence, if p<.5
and equation 4 holds (method 2 is better than nothing), then the rhs
of equation 3 is positive and method 2 is also better than method 1.
So if p<.5, equation 4 is a necessary and sufficient condition for
the offender committing the crime by method 2. Hence, if we wish to
prevent the crime, we must do it by making the rhs of equation 4
negative. Looking at the equation, we observe (in the final bracket
on the right) that Fi enters with a weight of p/(1-p)
relative to Fp. If p<.5, p/(1-p)<1, hence punishing an
impossible attempt provides less deterrence than an equal punishment
of a possible attempt, but still provides some deterrence.

Suppose, on the other hand, that p>.5: the offender believes
voodoo is more likely to work than poison. Looking at equation 3, we
observe that Method 2 is preferred if and only if

[ps(V-pcFc) -
pa(Fp+Fi)]<0. Eqn. 3'

Looking at Equation 4', we observe that, if 3' holds, then method
2 has a negative net return. Hence, if p>.5, the criminal either
uses voodoo or nothing.